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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and chemistry , Bragg's law, Wulff–Bragg's condition or Laue–Bragg interference, a special case of
Laue diffraction X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles ...
, gives the angles for coherent
scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...
of waves from a crystal lattice. It encompasses the superposition of wave fronts scattered by lattice planes, leading to a strict relation between wavelength and scattering angle, or else to the wavevector transfer with respect to the crystal lattice. Such law had initially been formulated for X-rays upon crystals. However, It applies to all sorts of quantum beams, including neutron and electron waves at atomic distances, as well as visible light at artificial periodic microscale lattices.


History

Bragg diffraction (also referred to as the Bragg formulation of X-ray diffraction) was first proposed by
Lawrence Bragg Sir William Lawrence Bragg, (31 March 1890 – 1 July 1971) was an Australian-born British physicist and X-ray crystallography, X-ray crystallographer, discoverer (1912) of Bragg's law, Bragg's law of X-ray diffraction, which is basic for t ...
and his father,
William Henry Bragg Sir William Henry Bragg (2 July 1862 – 12 March 1942) was an English physicist, chemist, mathematician, and active sportsman who uniquelyThis is still a unique accomplishment, because no other parent-child combination has yet shared a Nobel ...
, in 1913 in response to their discovery that
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
line solids produced surprising patterns of reflected
X-rays An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 Picometre, picometers to 10 Nanometre, nanometers, corresponding to frequency, ...
(in contrast to that of, say, a liquid). They found that these crystals, at certain specific wavelengths and incident angles, produced intense peaks of reflected radiation. The derived Bragg's law is a special interpretation of Laue diffraction, where the Braggs interpreted the constructive Laue-Bragg interference in a geometric way by reflection of waves from crystal lattice planes, such that the path-difference becomes a multiple of the incident wavelength. Lawrence Bragg explained this result by modeling the crystal as a set of discrete parallel planes separated by a constant parameter . It was proposed that the incident X-ray radiation would produce a Bragg peak if their reflections off the various planes interfered constructively. The interference is constructive when the phase shift is a multiple of ; this condition can be expressed by Bragg's law (see ''Bragg condition'' section below) and was first presented by Lawrence Bragg on 11 November 1912 to the
Cambridge Philosophical Society The Cambridge Philosophical Society (CPS) is a scientific society at the University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of law ...
. Although simple, Bragg's law confirmed the existence of real
particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object which can be described by several physical property, physical or chemical property, chemical ...
s at the atomic scale, as well as providing a powerful new tool for studying
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
s in the form of X-ray and neutron diffraction. Lawrence Bragg and his father, William Henry Bragg, were awarded the
Nobel Prize The Nobel Prizes ( ; sv, Nobelpriset ; no, Nobelprisen ) are five separate prizes that, according to Alfred Nobel's will of 1895, are awarded to "those who, during the preceding year, have conferred the greatest benefit to humankind." Alfr ...
in physics in 1915 for their work in determining crystal structures beginning with
NaCl Sodium chloride , commonly known as salt (although sea salt also contains other chemical salts), is an ionic compound with the chemical formula NaCl, representing a 1:1 ratio of sodium and chloride ions. With molar masses of 22.99 and 35.45 g ...
,
ZnS ZNS or ZnS may refer to: * Zinc sulfide or zinc sulphide (ZnS), a chemical compound * ZNS-TV, a national radio and TV broadcaster operated by the state-owned Broadcasting Corporation of The Bahamas ** ZNS-1 ZNS-1 (branded as ''Radio Bahamas'') i ...
, and
diamond Diamond is a Allotropes of carbon, solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Another solid form of carbon known as graphite is the Chemical stability, chemically stable form of car ...
. They are the only father-son team to jointly win. Lawrence Bragg was 25 years old, making him the youngest physics Nobel laureate. The concept of Bragg diffraction applies equally to
neutron diffraction Neutron diffraction or elastic neutron scattering is the application of neutron scattering to the determination of the atomic and/or magnetic structure of a material. A sample to be examined is placed in a beam of thermal or cold neutrons to o ...
and
electron diffraction Electron diffraction refers to the bending of electron beams around atomic structures. This behaviour, typical for waves, is applicable to electrons due to the wave–particle duality stating that electrons behave as both particles and waves. Si ...
processes. Both
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
and X-ray wavelengths are comparable with inter-atomic distances (~ 150 pm) and thus are an excellent probe for this
length scale In physics, length scale is a particular length or distance determined with the precision of at most a few orders of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot aff ...
.


Bragg condition

Bragg diffraction occurs when radiation of
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
comparable to atomic spacings, is scattered in a
specular Specular reflection, or regular reflection, is the mirror-like reflection of waves, such as light, from a surface. The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surfac ...
fashion (mirror-like reflection) by the atoms of a crystalline system, and undergoes constructive interference. For a crystalline solid, the waves are scattered from lattice planes separated by the distance between successive layers of atoms. When the scattered waves interfere constructively they remain in phase. They are reflected only when they strike the surface at a definite angle, the
glancing angle (optics) The angle of incidence, in geometric optics, is the angle between a incident ray, ray incident on a surface and the line perpendicular (at 90 degree angle) to the surface at the point of incidence, called the surface normal, normal. The ray can be f ...
(see figure on the right, and note that this differs from the convention in
Snell's law Snell's law (also known as Snell–Descartes law and ibn-Sahl law and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through ...
where is measured from the surface normal), the wavelength , and the "grating constant" of the crystal being connected by the relation: n\lambda = 2 d\sin\theta n is the
diffraction order In optics, a diffraction grating is an optical component with a periodic structure that diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form of structura ...
( n = 1 is first order, n = 2 is second order, n = 3 is third order). The effect of the constructive or destructive interference intensifies because of the cumulative effect of reflection in successive
crystallographic planes Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The word ...
(''h'',''k'',''l'') of the crystalline lattice (as described by Miller notation). This leads to Bragg's law, which describes the condition on ''θ'' for the constructive interference to be at its strongest: Note that moving particles, including
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
s,
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
s and
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
s, have an associated wavelength called ''
de Broglie wavelength Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave ...
''. A diffraction pattern is obtained by measuring the intensity of scattered waves as a function of scattering angle. Very strong intensities known as Bragg peaks are obtained in the diffraction pattern at the points where the scattering angles satisfy Bragg condition. As mentioned in the introduction, this condition is a special case of the more general
Laue equations In crystallography and solid state physics, the Laue equations relate incoming waves to outgoing waves in the process of elastic scattering, where the photon energy or light temporal frequency does not change by scattering, by a crystal lattice. T ...
, and the Laue equations can be shown to reduce to the Bragg condition under additional assumptions. The phenomenon of Bragg diffraction by a crystal lattice shares similar characteristics with that of
thin film interference Thin-film interference is a natural phenomenon in which light waves reflected by the upper and lower boundaries of a thin film interfere with one another, either enhancing or reducing the reflected light. When the thickness of the film is an ...
, which has an identical condition in the limit where the refractive indices of the surrounding medium (e.g. air) and the interfering medium (e.g. oil) are equal.


Underpinning scattering processes

When
X-ray An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10  picometers to 10  nanometers, corresponding to frequencies in the range 30&nb ...
s are incident on an
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, and ...
, they make the electronic cloud move, as does any
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
. The
movement Movement may refer to: Common uses * Movement (clockwork), the internal mechanism of a timepiece * Motion, commonly referred to as movement Arts, entertainment, and media Literature * "Movement" (short story), a short story by Nancy Fu ...
of these
charges Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * '' Charge!!'', an album by The Aqu ...
re-radiates
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...
s with the same
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, blurred slightly due to a variety of effects; this phenomenon is known as
Rayleigh scattering Rayleigh scattering ( ), named after the 19th-century British physicist Lord Rayleigh (John William Strutt), is the predominantly elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the ...
(or elastic scattering). The scattered waves can themselves be scattered but this secondary scattering is assumed to be negligible. A similar process occurs upon scattering neutron waves from the nuclei or by a
coherent Coherence, coherency, or coherent may refer to the following: Physics * Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference * Coherence (units of measurement), a deri ...
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
interaction with an unpaired
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
. These re-emitted wave fields interfere with each other either constructively or destructively (overlapping waves either add up together to produce stronger peaks or are subtracted from each other to some degree), producing a diffraction pattern on a detector or film. The resulting wave interference
pattern A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated l ...
is the basis of
diffraction Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
analysis. This analysis is called ''Bragg diffraction''.


Heuristic derivation

Suppose that a single
monochromatic A monochrome or monochromatic image, object or color scheme, palette is composed of one color (or lightness, values of one color). Images using only Tint, shade and tone, shades of grey are called grayscale (typically digital) or Black and wh ...
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...
(of any type) is incident on aligned planes of
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
points, with separation d, at angle \theta. Points A and C are on one plane, and B is on the plane below. Points ABCC' form a
quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
. There will be a path difference between the ray that gets reflected along AC' and the ray that gets transmitted along AB, then reflected along BC. This path difference is (AB + BC) - \left(AC'\right) \,. The two separate waves will arrive at a point (infinitely displaced from these lattice planes) with the same
phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform * Phase space, a mathematic ...
, and hence undergo
constructive interference In physics, interference is a phenomenon in which two waves combine by adding their displacement together at every single point in space and time, to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive ...
, if and only if this path difference is equal to any integer value of the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
, i.e. n\lambda =(AB + BC) - \left(AC'\right) where n and \lambda are an integer and the wavelength of the incident wave respectively. Therefore, AB = BC = \frac \text AC = \frac \,, from which it follows that AC' = AC\cdot\cos\theta = \frac\cos\theta = \left(\frac\cos\theta\right)\cos\theta = \frac\cos^2\theta \,. Putting everything together, n\lambda = \frac - \frac\cos^2\theta = \frac\left(1 - \cos^2\theta\right) = \frac\sin^2\theta which simplifies to n\lambda = 2d\sin\theta \,, which is Bragg's law shown above. If only two planes of atoms were diffracting, as shown in the pictures, then the transition from constructive to destructive interference would be gradual as a function of angle, with gentle maxima at the Bragg angles. However, since many atomic planes are participating interference in most real materials, very sharp peaks surrounded by mostly destructive interference result. A rigorous derivation from the more general Laue equations is available (see page:
Laue equations In crystallography and solid state physics, the Laue equations relate incoming waves to outgoing waves in the process of elastic scattering, where the photon energy or light temporal frequency does not change by scattering, by a crystal lattice. T ...
).


Bragg scattering of visible light by colloids

A
colloidal crystal A colloidal crystal is an ordered array of colloid particles and fine grained materials analogous to a standard crystal whose repeating subunits are atoms or molecules. A natural example of this phenomenon can be found in the gem opal, where sphere ...
is a highly ordered array of particles that forms over a long range (from a few
millimeters 330px, Different lengths as in respect to the electromagnetic spectrum, measured by the metre and its derived scales. The microwave is between 1 meter to 1 millimeter. The millimetre (American and British English spelling differences#-re, -er, ...
to one
centimeter 330px, Different lengths as in respect to the Electromagnetic spectrum, measured by the Metre and its deriveds scales. The Microwave are in-between 1 meter to 1 millimeter. A centimetre (international spelling) or centimeter (American spellin ...
in length); colloidal crystals have appearance and properties roughly
analogous Analogy (from Greek ''analogia'', "proportion", from ''ana-'' "upon, according to" lso "against", "anew"+ ''logos'' "ratio" lso "word, speech, reckoning" is a cognitive process of transferring information or meaning from a particular subject ( ...
to their atomic or molecular counterparts. It has been known for many years that, due to repulsive
Coulombic Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
interactions,
electrically charged Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
macromolecules A macromolecule is a very large molecule important to biophysical processes, such as a protein or nucleic acid. It is composed of thousands of covalently bonded atoms. Many macromolecules are polymers of smaller molecules called monomers. The ...
in an
aqueous An aqueous solution is a solution in which the solvent is water. It is mostly shown in chemical equations by appending (aq) to the relevant chemical formula. For example, a solution of table salt, or sodium chloride (NaCl), in water would be rep ...
environment can exhibit long-range
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
-like correlations, with interparticle separation distances often being considerably greater than the individual particle diameter. Periodic arrays of spherical particles give rise to interstitial voids (the spaces between the particles), which act as a natural
diffraction grating In optics, a diffraction grating is an optical component with a periodic structure that diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form of structura ...
for visible light waves, when the interstitial spacing is of the same
order of magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic dis ...
as the
incident Incident may refer to: * A property of a graph in graph theory * ''Incident'' (film), a 1948 film noir * Incident (festival), a cultural festival of The National Institute of Technology in Surathkal, Karnataka, India * Incident (Scientology), a ...
lightwave. In these cases in nature, brilliant
iridescence Iridescence (also known as goniochromism) is the phenomenon of certain surfaces that appear to gradually change color as the angle of view or the angle of illumination changes. Examples of iridescence include soap bubbles, feathers, butterfl ...
(or play of colours) is attributed to the diffraction and
constructive interference In physics, interference is a phenomenon in which two waves combine by adding their displacement together at every single point in space and time, to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive ...
of visible lightwaves according to Bragg's law, in a matter analogous to the
scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...
of
X-rays An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 Picometre, picometers to 10 Nanometre, nanometers, corresponding to frequency, ...
in crystalline solid. The effects occur at visible wavelengths because the separation parameter is much larger than for true crystals. Precious
opal Opal is a hydrated amorphous form of silica (SiO2·''n''H2O); its water content may range from 3 to 21% by weight, but is usually between 6 and 10%. Due to its amorphous property, it is classified as a mineraloid, unlike crystalline forms ...
is one example of colloidal crystals producing striking optical effects.


Volume Bragg gratings

Volume Bragg gratings (VBG) or volume holographic gratings (VHG) consist of a volume where there is a periodic change in the
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
. Depending on the orientation of the modulation of the refractive index, VBG can be used either to
transmit Transmit is a file transfer client program for macOS. Developed by Panic, Transmit is shareware. After a seven-day trial period, the product can only be used for seven-minute sessions until it has been purchased. Originally built as an FTP client ...
or reflect a small bandwidth of
wavelengths In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
. Bragg's law (adapted for volume hologram) dictates which wavelength will be diffracted: 2\Lambda\sin(\theta + \varphi)=m\lambda_B \,, where is the Bragg order (a positive integer), the diffracted
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
, Λ the fringe spacing of the grating, the angle between the incident beam and the normal () of the entrance surface and the angle between the normal and the grating vector (). Radiation that does not match Bragg's law will pass through the VBG undiffracted. The output wavelength can be tuned over a few hundred nanometers by changing the incident angle (). VBG are being used to produce widely tunable laser source or perform global hyperspectral imagery (see Photon etc.).


Selection rules and practical crystallography

Bragg's law, as stated above, can be used to obtain the lattice spacing of a particular cubic system through the following relation: d = \frac \,, where a is the lattice spacing of the
cubic crystal In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of ...
, and , , and are the
Miller indices Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices. In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers ''h'', ''k'', and ''â ...
of the Bragg plane. Combining this relation with Bragg's law gives: \left(\frac\right)^2 = \left(\frac\right)^2 \frac One can derive selection rules for the
Miller indices Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices. In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers ''h'', ''k'', and ''â ...
for different cubic Bravais lattices; here, selection rules for several will be given as is. These selection rules can be used for any crystal with the given crystal structure. KCl has a face-centered cubic
Bravais lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n_ ...
. However, the K+ and the Cl− ion have the same number of electrons and are quite close in size, so that the diffraction pattern becomes essentially the same as for a simple cubic structure with half the lattice parameter. Selection rules for other structures can be referenced elsewhere, or
derived Derive may refer to: * Derive (computer algebra system), a commercial system made by Texas Instruments * ''Dérive'' (magazine), an Austrian science magazine on urbanism *Dérive, a psychogeographical concept See also * *Derivation (disambiguatio ...
. Lattice spacing for the other
crystal system In crystallography, a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of Bravais lattices. Space groups are classified into crystal systems according to their po ...
s can be found
here Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Television * Here TV (formerly "here!"), a TV ...
.


See also

*
Bragg plane In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, \scriptstyle \mathbf, at right angles. The Bragg plane is defined as part of the Von Laue condition for diffraction peaks in x-ray diffraction c ...
*
Crystal lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
*
Diffraction Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
*
Distributed Bragg reflector A distributed Bragg reflector (DBR) is a reflector used in waveguides, such as optical fibers. It is a structure formed from multiple layers of alternating materials with varying refractive index, or by periodic variation of some characteristic ...
**
Fiber Bragg grating A fiber Bragg grating (FBG) is a type of distributed Bragg reflector constructed in a short segment of optical fiber that reflects particular wavelengths of light and transmits all others. This is achieved by creating a periodic variation in th ...
*
Dynamical theory of diffraction The dynamical theory of diffraction describes the interaction of waves with a regular lattice. The wave fields traditionally described are X-rays, neutrons or electrons and the regular lattice are atomic crystal structures or nanometer-scale multi ...
* Georg Wulff *
Henderson limit The Henderson limit is the X-ray dose (energy per unit mass) a Cryogenics, cryo-cooled crystal can absorb before the diffraction pattern decays to half of its original intensity. Its value is defined as 2 × 107 Gray (unit), Gy (J/kg). Decay of ...
* Laue conditions *
Powder diffraction Powder diffraction is a scientific technique using X-ray, neutron, or electron diffraction on powder or microcrystalline samples for structural characterization of materials. An instrument dedicated to performing such powder measurements is call ...
* Radar angels *
Structure factor In condensed matter physics and crystallography, the static structure factor (or structure factor for short) is a mathematical description of how a material scatters incident radiation. The structure factor is a critical tool in the interpretation ...
*
X-ray crystallography X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles ...


References


Further reading

* Neil W. Ashcroft and N. David Mermin, ''Solid State Physics'' (Harcourt: Orlando, 1976). *


External links


Nobel Prize in Physics – 1915
* https://web.archive.org/web/20110608141639/http://www.physics.uoguelph.ca/~detong/phys3510_4500/xray.pdf

{{DEFAULTSORT:Bragg's Law Diffraction Neutron X-rays Crystallography